Answer
The graph is symmetric with respect to the $y$-axis, but not with respect to the $x$-axis and origin.
Work Step by Step
$y = |x^{3} + x|$
The graph of an equation in $x$ and $y$ is symmetric with respect to the $y$-axis
when replacing $x$ by $-x$ yields an equivalent equation.
$y = |(-x)^{3} - x|$
The part of the equation between vertical bars is an absolute value, meaning that only if it returns a negative, it will be multiplied by $-1$.
$(-x)^{3} - x$ always gives the exact negative result of $x^{3} + x$ meaning it has the same absolute value so the equation remains the same.
The graph of an equation in $x$ and $y$ is symmetric with respect to the $x$-axis
when replacing $y$ by $-y$ yields an equivalent equation.
When $y$ becomes $-y$ nothing else changes in the equation so it is not the same as the original.
The graph of an equation in $x$ and $y$ is symmetric with respect to the origin
when replacing $x$ by $-x$ and $y$ by $-y$ yields an equivalent equation.
Because the right member of the equation remains unchanged when replacing $x$ by $-x$ but the left does when replacing $y$ by $-y$ the resulting equation is not the same as the original.
